---
id: "2026-10-06-openai-unit-distance-upper-bound-power-saving-claim"
url: "https://postcutoff.com/e/2026-10-06-openai-unit-distance-upper-bound-power-saving-claim/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Peer-reviewed
sources: 8
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-unit-distance-upper-bound-power-saving-claim/

# Unit-distance bound below n^(4/3) claimed (OpenAI, Lean)

Full title: OpenAI's math release claims the first improvement on the Spencer–Szemerédi–Trotter bound of 1984: n points in the plane span O(n^(4/3−δ)) unit distances; also the weak pinned distinct-distance conjecture; both in Lean, and Will Sawin has explained a route to an effective proof

"A power saving for planar unit distances" (OpenAI, 23 Sept 2026, 53 pages; family 167 of the 6 Oct release): "We prove a power saving for the planar unit-distance problem: for some absolute β < 4/3, every set of n points in the Euclidean plane determines O(n^β) unordered pairs at unit distance." The companion "The weak pinned planar distance theorem" (27 pages) proves that, for every ε > 0, all but o(n) points of an n-point planar set determine at least n^(1−ε) distinct distances. Both are formalized in Lean (PlanarUnitDistances, PinnedDistances) and were accepted in the oai-math-revised comparator run. Thomas Bloom: the PDF "gives no indication as to the value of c obtained" and combines the point–circle incidence view with "some kind of entropy-based argument together with input from algebraic number theory". teorth/optimizationproblems merged the 4/3 − ε upper bound on 18 Oct.

## Key facts
- Manuscript (abstract, verbatim): 'We prove a power saving for the planar unit-distance problem: for some absolute β < 4/3, every set of n points in the Euclidean plane determines O(n^β) unordered pairs at unit distance.' Theorem 1.1: absolute constants 0 < C < ∞ and 1 ≤ β < 4/3 with u(n) ≤ C·n^β for every n. 53 pages, dated 23 Sept 2026
- Background (paper): Erdős posed the problem in 1946 with the lattice lower bound n^(1+c/log log n); Spencer, Szemerédi and Trotter proved u(n) = O(n^(4/3)) in 1984, reproved by Székely's crossing-number method; Pach, Raz and Solymosi obtained a logarithmic improvement conditional on a rigidity conjecture. In May 2026 an OpenAI construction disproved Erdős's conjecture u(n) = n^(1+o(1)) (see [2026-05-20-ai-disproves-erdos-unit-distance-conjecture](https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/index.md)); Sawin sharpened it to n^1.014114, and later work pushed the lower exponent to about 1.043
- Companion (abstract, verbatim): 'We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but o(n) points of any n-point planar set determine at least n^(1−ε) distinct nonzero distances.' 27 pages; Bloom lists it under Erdős problem #604
- Lean (OpenAI scope note lean/docs/167.md): 'there are absolute constants C>0 and 1≤β<4/3 such that every finite planar point set of size n determines at most Cn^β unordered unit-distance pairs. The same constants work for every n'; the pinned-distance statement is a separate challenge. Both 'accepted' in the oai-math-revised comparator run of 25–26 Oct
- Thomas Bloom (erdosproblems.com, problem #90, 8 Oct): 'The best lower bound for this problem, obtained by OpenAI in May, is n^{1+c} for some constant c>0. The best upper bound previously, due to Spencer, Szemerédi, and Trotter [SST84], was n^{4/3}. The PDF gives no indication as to the value of c obtained.' The proof 'appears to begin with a point-circle incidence interpretation (which was used in [SST84]), and couples that with some kind of entropy-based argument together with input from algebraic number theory'
- Expert reading (forum comment by WillSawin, 25 Oct): 'The biggest progress is that I see how to remove the argument with ultrafilters from section 8, which was one of the things making the argument ineffective, and replace it with in-principle effective algebraic geometry reasoning.' He describes the strategy as extracting structure from near-extremal configurations of the n^(4/3) argument and deriving a contradiction 'in a somewhat surprising way', with a key role for the Weil height of ratios on unit circles, and says of the entropy step: 'I don't understand this'
- Recognition of the bound: teorth/optimizationproblems PR #204 (merged by teorth on 18 Oct) adds 'OpenAI, A power saving for planar unit distances' as the source of the upper exponent 4/3 − ε with ε not explicit, and corrects 'The comment saying the exponent 4/3 had not been improved'. Gil Kalai's 7 Oct post lists the improved planar unit-distance bound among the release's results

## What happened

Erdős asked in 1946 how many pairs among n points in the plane can be at distance exactly one. Spencer, Szemerédi and Trotter proved
the upper bound O(n^(4/3)) in 1984 from incidences between points and unit circles. Later work improved only the constant. In
May 2026 an OpenAI model refuted Erdős's conjecture that the truth is n^(1+o(1)), with a number-field construction giving n^(1+c).

OpenAI's 6 Oct release now claims the matching improvement from above: O(n^(4/3−δ)) for an absolute δ that the paper does not compute.
A companion paper proves the weak pinned distance conjecture. Both are formalized in Lean, and a community comparator run accepted them.
On 25 Oct a commenter posting as WillSawin, the name of the mathematician who sharpened the May construction, posted a long reading on
erdosproblems.com. He traces the argument to near-extremal incidence structure and Weil heights of ratios of points on unit circles,
and says he can replace its ultrafilter step with effective algebraic geometry.

## Why it matters

If correct, the claim breaks a 42-year-old barrier in incidence geometry. Combined with May's lower bound, the unit-distance exponent
lies strictly between 1 and 4/3, and both moves came from OpenAI models using algebraic number theory. A detailed outside reading and
the formal proof make this one of the better-checked claims in the release, but δ is still unknown and the proof is not refereed.

## What is disputed or not yet verified

- Verification: Lean formalizations accepted by an independent comparator run; unrefereed manuscripts; an outside expert's partial reading proposes an effective replacement for one step

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI (PDF): A power saving for planar unit distances (23 Sept 2026)](https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf) (github.com, paper)
2. [OpenAI (PDF): The weak pinned planar distance theorem (23 Sept 2026)](https://github.com/openai/math/blob/main/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf) (github.com, paper)
3. [OpenAI Lean scope note for family 167](https://github.com/openai/math/blob/main/lean/docs/167.md) (github.com, code)
4. [teorth/optimizationproblems PR #204: unit distance bounds, upper bound 4/3 − ε (merged 18 Oct)](https://github.com/teorth/optimizationproblems/pull/204) (github.com, code)
5. [oai-math-revised comparator results (PlanarUnitDistances, PinnedDistances accepted)](https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv) (github.com, code)
6. [Thomas Bloom: Erdős problems in the OpenAI proofs (problems #90, #604) and WillSawin's comment of 25 Oct](https://www.erdosproblems.com/forum/thread/blog:10) (erdosproblems.com, discussion)
7. [Gil Kalai: Updates: Sharing AI progress on mathematics (amazing!) (7 Oct)](https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/) (gilkalai.wordpress.com, discussion)
8. [oai-math-revised AGENT_NOTES for the unit-distance manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29 (filed): Created (CONTENTS entry, both PDFs (53 and 27 pp), Lean scope note, comparator results, Bloom's post and the WillSawin comment, teorth PR #204, Kalai)

## Related

- 2026-10-24: [oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)](https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/index.md)
- 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)
- 2026-10-06: [Erdős's arithmetic-progression conjecture claimed (OpenAI, Lean)](https://postcutoff.com/e/2026-10-06-openai-erdos-arithmetic-progressions-conjecture-claim/index.md)
- 2026-05-20: [OpenAI model disproves Erdős's 80-year-old unit distance conjecture](https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/index.md)
- People: [Thomas Bloom](https://postcutoff.com/person/thomas-bloom/)